Spin-orbit coupling in wurtzite heterostructures

A detailed derivation of the Rashba spin-orbit (SO) Hamiltonian for conduction electrons in wurtzite heterostructures is lacking in the literature. Here we derive in a consistent and systematic way such an effective Hamiltonian, valid for quantum wells, wires, and dots with arbitrary confining potentials and external magnetic fields. We start from an 8 x 8 Kane model accounting for the s-p(z) orbital mixing important to wurtzite structures, but absent in zincblende, and apply both quasidegenerate perturbation theory (Lowdin partitioning) and the folding down approach to derive an effective 2 x 2 Hamiltonian for conduction electrons. For bulk systems, our derivation consistently yields the well-known linear-in-momentum bulk inversion asymmetry (BIA) Rashba-like term, with SO coupling alpha(BIA)(bulk), entirely following from the s-p(z) orbital mixing and in agreement with experiments. We also obtain the correct form of the bulk Dresselhaus term, which is the same as that of the Rashba. However, our calculated bulk Dresselhaus SO parameters gamma and b are too small. Focusing on wurtzite quantum wells, we perform a self-consistent Poisson-Schrodinger calculation in the Hartree approximation to determine all the relevant SO couplings of the confined effective 2 x 2 electron Hamiltonian. Our total linear Rashba-type SO Hamiltonian contains a structural inversion asymmetry (SIA) part, modulated by the Hartree, doping, and external gate potentials of the wells, and, in contrast to zincblende structures, a confined Rashba-type contribution induced by the BIA of the underlying wurtzite lattice. Our calculation shows this latter BIA term to be the main contribution to the confined Rashba coupling in wurtzite wells. As a concrete example, we determine the intrasubband (intersubband) Rashba alpha(upsilon) (eta) and linear Dresselhaus beta(upsilon) (Gamma) SO coupling strengths for GaN/AlGaN single and double wells with one and two occupied subbands (upsilon = 1, 2). Since the linear Rashba and the Dresselhaus terms have the same functional form, we can define a total effective SO coupling alpha(eff)(upsilon)=alpha(upsilon)+beta(upsilon). For the GaN/Al0.3Ga0.7N single well with one occupied subband we find alpha(eff )(1)= 7.16 meV angstrom, in agreement with weak antilocalization measurements. In the case of two occupied subbands, we observe that the intersubband Rashba eta is much weaker than the intrasubband coupling alpha(upsilon). For double wells even in the presence of strong built-in electric fields (spontaneous and piezoelectric, crucial in GaN/AlGaN wells), we find a seemingly symmetric potential configuration at which both the Rashba eta and Dresselhaus F intersubband couplings exhibit their highest strengths. On the other hand, we observe that the intrasubband Dresselhaus couplings beta(1) and beta(2) interchange their values as the gate voltage V-g varies across zero; a similar behavior, though less pronounced, is seen for the Rashba couplings alpha(1) and alpha(2) . We believe our general effective Hamiltonian for electrons in wurtzite heterostructures put forward here, should stimulate additional theoretical works on wurtzite quantum wells, wires, and dots with variously defined geometries and external magnetic fields.